Let's tackle the problem step-by-step.
1. Understanding the Rates:
- Bethany can mow the lawn in 4 hours. This means her rate is [tex]\(\frac{1}{4}\)[/tex] lawns per hour.
- Colin can mow the lawn in 3 hours. This means his rate is [tex]\(\frac{1}{3}\)[/tex] lawns per hour.
2. Combining the Rates:
When they work together, their combined rate is the sum of their individual rates. So, we add their rates together:
[tex]\[\text{Combined rate} = \frac{1}{4} + \frac{1}{3}\][/tex]
3. Setting Up the Equation:
Since they are working together to mow the entire lawn, we let [tex]\(x\)[/tex] be the number of hours it takes for them to mow the lawn together. The fraction of the lawn that each can mow in [tex]\(x\)[/tex] hours would be:
- Bethany mows [tex]\(\frac{1}{4} x\)[/tex] of the lawn.
- Colin mows [tex]\(\frac{1}{3} x\)[/tex] of the lawn.
Together, they should finish mowing 1 entire lawn. Therefore, we set up the equation:
[tex]\[\frac{1}{4} x + \frac{1}{3} x = 1\][/tex]
4. Solving the Equation:
To solve this equation for [tex]\(x\)[/tex], you would find a common denominator, combine the terms, and isolate [tex]\(x\)[/tex]. From the detailed calculation already obtained, we know the combined rate and the solution for [tex]\(x\)[/tex]:
[tex]\[\frac{1}{4} + \frac{1}{3} = 0.5833333333333333\][/tex]
So, [tex]\(x\)[/tex] would be:
[tex]\[x = \frac{1}{0.5833333333333333} = 1.7142857142857144\][/tex]
Therefore, it would take Bethany and Colin approximately 1.714 hours to mow the lawn together. The correct equation to find the number of hours [tex]\(x\)[/tex] it would take them to mow the lawn together is:
[tex]\[\boxed{\frac{1}{4} x + \frac{1}{3} x = 1}\][/tex]
